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Let $P$ be a nonconstant polynomial with constant term equal $0$ such that $\lim_{n\to\infty}\sin(P(n)\pi)$ and $\lim_{n\to\infty}\cos(P(n)\pi)$ exist.

Prove or disprove: all coefficients of $P$ are rational.

The problem is based on my own investigations. Related. Both might be known, but I couldn't find results of that sort.

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larry01
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1 Answers1

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Yes, this is true. The following holds:

Theorem: Let $P(x) = a_nx^n+...+a_0$ be a polynomial such that for some $j>0$, $a_j$ is irrational. Then $\sin(P(n)\pi)$ is a sequence dense in $[-1, 1]$.

Look at my previous answer as a reference.

Jakobian
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